# Systematics of toroidal dipole modes in Ca, Ni, Zr, and Sn isotopes

## TL;DR

This study uses self-consistent QRPA calculations to analyze the relationship between toroidal dipole modes and pygmy dipole resonance in various isotopes, revealing that low-lying dipole strength is mainly toroidal motion.

## Contribution

It demonstrates that low-lying dipole strength traditionally attributed to PDR is actually predominantly toroidal flow, providing a new interpretation of these excitations.

## Key findings

- Toroidal flow pattern is dominant in low-energy dipole excitations.
- A one-to-one correspondence exists between transition densities and currents.
- Toroidal motion is present even in uncoupled two-quasiparticle excitations.

## Abstract

We analyze the relation between isoscalar toroidal modes and so-called pygmy dipole resonance (PDR) which both appear in the same region of low-energy dipole excitations. To this end, we use a theoretical description within the fully self-consistent Skyrme quasiparticle random-phase approximation (QRPA). Test cases are spherical nuclei $^{40,48}$Ca, $^{58,72}$Ni, $^{90,100}$Zr, and $^{100,120,132}$Sn which cover four different elements and for each element at least two isotopes with different neutron excess, one small and another large. The structure of the modes is investigated in terms of strength functions, transition densities (TD) and transition currents (TC). For all considered nuclei, we see that, independently on whether PDR strength exists or not, the flow pattern in the lower part of the "PDR energy region" is basically the isoscalar vortical toroidal motion with a minor irrotational fraction. A one-to-one correspondence between calculated TD and TC can be established. The toroidal flow appears already in the uncoupled two-quasiparticle (2qp) excitations and becomes definitively strong for the QRPA modes. Altogether, we find that low-lying dipole strength often denoted as isoscalar PDR is actually an oversimplified imitation of the basically toroidal motion in nuclei with a sufficient neutron excess.